Discrete Data

Discrete Data (Attribute Data) - represents counted / classified / categorized data. Only a finite number of values are possible and it cannot be subdivided meaningfully. E.g. number of defects, number of people in a room, number of products audited etc.

An application oriented question on the topic along with responses can be seen below. The best answer was provided by Kavitha Sundar on 6th October 2017.

## Question

Q4 in Episode 2 - While continuous data is measured and attribute data is counted, there is sometimes confusion if some specific dataset should be considered continuous or attribute. Provide some examples of confusing datasets and your inference.

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

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Question 4 in Episode 2:

While continuous data is measured and attribute data is counted, there is sometimes confusion if some specific dataset should be considered continuous or attribute. Provide some examples of confusing datasets and your inference.

Data – is defined as a collection of avalues / useful information that is required for any analysis to the receipient. Data is genereally used to prove / disprove hypothesis.

Data is of two types basis statistics. It is Quantitative or Qualitative.

Quantitative is descriptive data, which can be categorized into subgroups for analysis and qualitative is numerical which means either measurable / countable. Qualitative data is again divided into 2 types continuous and discrete data.

For Eg.

Charlie chaplin is fair, short, has small mustache, thin built and wears black colored jacket. – it is qualitative data.

Charlie chaplin has one hat, one walking stick and 2 legs. – it is Quantitative –discrete data.

Charlie chaplin aged 45 years is 57.2 kgs built and 4.8 inches tall . – it is quantitative continuous data.

4 types of measurement scales:

It is divided into four categories – Nominal and ordinal, interval and ratio

Ø  Nominal data: It assigns a numerical value as an attribute to any object / animal / person / any non-numerical data.

Ø  Ordinal data: Any data which can be ordered and ranked is called ordinal data. This can’t be measured.

Eg. 1. A horse is numbered in the race court, represents the nominal data.

2.       The numbered winning horses are ordered and ranked as “1st, 2nd  and 3rd place” in race club, which represents ordinal data. Another best examples is progress report of the student.

Ø  Interval: It is a numeric scale where we know order as well as the differences between values. There is no origin.

Eg. Temperature of the room is set to be normal if it is between 25 and 28 degrees C. Time is another good example of an interval scale in which the increments are known, consistent, and measurable.

Ø  Ratio: Ratio scales are the ultimate nirvana when it comes to measurement scales because they tell us about the order, they tell us the exact value between units, AND they also have an absolute zero–which allows for a wide range of both descriptive and inferential statistics to be applied.  At the risk of repeating myself, everything above about interval data applies to ratio scales + ratio scales have a clear definition of zero.

Good examples of ratio variables include height and weight.

Qualitative data:

It is otherwise called as categorical data.

Quantitative data:

It is divided into two contionus and discrete data.

Difference between Continuous and discrete data:

 Continuous data Discrete data It is measureable on a scale It is countable The data falls within finite or infinite range The data has only finite numbers. Can be broken into subcategories Can't be broken since it is a whole number. The frequency is depicted in histogram, where skewness is shown clearly the values take a distinct value hence it is represented in bar diagram, skewness can't be seen. Values are allowed to group within the range The values are individual values. Eg. Temperature of the person, Height, Weight, Age, time, Cycle time taken to complete a task Eg. No. of cumputeers, No. of students, no. of books, no. of certificates, no. of errors, etc

Confusion between Contionus and Discrete data:

Eg. 1:

 Person Age Weight (Kgs) Height(Inches) Color Ajay 34 51 5.1 Wheatish Sharma 35 65.5 5.2 Fair Roshini 23 45.5 4.8 Wheatish Gaithri 53 72.5 4.8 Dark Linda 43 46.5 5.1 Fair Tanya 36 43 5.3 Wheatish Balu 27 56 5.6 Fair Vignesh 32 77 6.1 Dark Aarav 43 76 5.9 Wheatish Rithesh 45 64 5.3 Dark

Qualitative data / categorical data:

Categorize 10 people in the group into wheatish, dark, fair basis the color. This represents categorical data.

Continuous data:

Age , Height and weight of the people displayed above in the table depicts a good example of continuous data, where these numbers falls within the infinite ranges.

Discrete data:

No . of Wheatish – 4

No. of fair – 3

No. of dark – 3

Total no. of people – 10

Conclusion of Eg. 1: Though age is continuous numerical variables. Although the recorded ages have been truncated to whole numbers, the concept of age is continuous.) Number of aged people is a discrete numerical variable (a count).

Age can be rounded down to a whole number, if so it represents the discrete data. Though it falls under discrete(when all data is shown as whole integers), it is actually a continuous data because it has ranges. Age is not a constant factor, though the DOB is constant.

Basis the context / concept of the requirement – lets say to fill a form, the exact age is required. In such case, though age is discrete, it is continuous.

“12 years, 153 days” really means a continuous age that is between 12Y152.5D and 12Y153.5D.

Eg. 2 : Income is another example of continuous data.

Eg. 3:

In practice, percentage data are often treated as continuous because thepercentage can take on any value along the continuum from zero to 100%. In addition, dividing a percentage point into two or more parts still makes sense.Discrete data are easy to collect and interpret.

% is always to be considered as continuous but it depends on the concept.

If I have to track the error percentage, the right metric is as below..

Error % =          No of errors (Discrete)

Total charts audited.(Discrete)

Hence Error % is discrete.

Another example:

If I have to track the availability of the machine, the formula is as follows…

Availability % = Total hours available (Continuous) / Expected hours of production for 8 hours(Continuous)

Hence Availability % is continuous, since time is continuous.

Conclusion:

It depends….  In certain situations, discrete data may take on characteristics of continuous data.  But, if counts are large, distribution of values are relatively wide, and the the values are distributed across the values, you can “pretend” it is continuous and use the appropriate tools.

Thanks

Kavitha

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First reference for confusion between continuous and attribute data is when there is mix of continuous and attribute measures that forms a metric. Example Defects density is a metric derived by dividing total number of defects by total effort spent in the project. Here total defects is a attribute data where as total effort is a continuous data, but since atleast one of them is continuous data, overall outcome which is defects density is considered as continuous data.

Other confusing scenario is pseudo continuous data, When you treat discrete data as continuous, it may be referred to as pseudo continuous. And when you take averages of ratings for let's say 10 bank tellers. An example - This could be a 0 to 10 rating scale. You may get an average rating of 7.5. One should usually manipulate discrete data into continuous or consider it as pseudo continuous with the intent of using tests applicable for continuous data. (Please note this was clarification provided by VK or Rupinder from Benchmark on one of my question).

Other confusing scenario is when we need to consider, data points above 10 as continuous else discrete.

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Continuous data is measured because it can take any value between two values. There is infinite possibility to measure reading between two readings.

For Example -

1) Time taken to process a part – it can be 1hr or 1.1hr or 1.123hr, depending upon how much detailing is essential for problem solving.

2) Temp, Pressure, Speed, Time, Length, Space, etc

Where as Attribute data is counted because there it takes only integer & finite values.

For example –

1) Number of defective cars in Sept-17, Car with even a single defect will also be counted as defective car, but there will not be fraction or infinite possibilities like 0.5car, 0.25car etc. Car is defective or not defective, thats all.

2) Number of pregnant women in a city; there cannot be half pregnant women. Its like pregnant or not pregnant thats all.

Confusion starts when we go in more details of defining attribute or its refinement. Practically it is possible to have infinite ranges of attributes,
For example -

1) No of persons who have passed SSC in a particular age group range, now here observer can note number of persons in range of age in years, like persons of 15 years old, 16 years old, 17 years old etc…. But practically any two persons will not have exact same age, there is definitely infinite differences, in terms of Months, days, hours, minutes, seconds, milliseconds, microseconds and so on (its infinite)…….  there is infinite possibility of making these “counting ranges” hence attribute data can be sometimes confused as continuous data in specific cases. In this example, it depends on problem statement, how much detailing is significant is to be decided by problem solver. Age in terms days is OK but in terms of hrs and beyond detailing will be insignificant and so one should group it in closest attribute range.

2) Other examples are, huge data counts or large amount of discrete entities like

• Plants
• biological data
• human population specifications data
• grains
• material quality in mining industry
• marine science data etc

We may not give significance to 50000000009 and 50000000007 and its difference, practically these are continuous but we may consider in to closest discrete range.

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Continuous data is measured and attribute data is counted.

There are some cases when we actually count our measurement, but for analysis purpose those data are considered continuous.

Below are some examples.

Example1:- In a 3 Kg packet of French Fry, we measure count per Kg. which is an attribute data as it is counted. But if we want to check what is the mean and median of count in a 3 Kg pack, we need to consider the same data as continuous.

 Short text for inspection object Count Per Kg Fries 3 Kg 374.000 Fries 3 Kg 372.000 Fries 3 Kg 376.000 Fries 3 Kg 379.000 Fries 3 Kg 372.000 Fries 3 Kg 372.000 Fries 3 Kg 375.000 Fries 3 Kg 358.000 Fries 3 Kg 369.000 Fries 3 Kg 368.000 Fries 3 Kg 367.000

Count Per Kg Quantiles

 100.0% maximum 386 99.5% 383 97.5% 382 90.0% 379 75.0% quartile 376 50.0% median 372 25.0% quartile 367 10.0% 358 2.5% 351 0.5% 347 0.0% minimum 344

Summary Statistics

 Mean 370.386 Std Dev 8.02324 Std Err Mean 0.369691 Upper 95% Mean 371.112 Lower 95% Mean 369.659 N 471 Sum 174452 Variance 64.3723 CV 2.16618 N Missing 476

Example 2:-

Similar to above example, for a sample of 1 Kg, we measure critical and major defects. Which again is an attribute data, but for analysis they are considered continuous data.

 Short text for inspection object Critical + Major + Minor Defects Fries 3 Kg 10.000 Fries 3 Kg 5.000 Fries 3 Kg 8.000 Fries 3 Kg 15.000 Fries 3 Kg 13.000 Fries 3 Kg 14.000 Fries 3 Kg 8.000 Fries 3 Kg 10.000 Fries 3 Kg 2.000 Fries 3 Kg 10.000 Fries 3 Kg 14.000 Fries 3 Kg 9.000 Fries 3 Kg 9.000 Fries 3 Kg 11.000 Fries 3 Kg 10.000 Fries 3 Kg 11.000 Fries 3 Kg 7.000 Fries 3 Kg 8.000 Fries 3 Kg 8.000 Fries 3 Kg 7.000

Distributions

Critical + Major Other Quantiles

 100.0% maximum 22 99.5% 19.175 97.5% 12.175 90.0% 8 75.0% quartile 6 50.0% median 4 25.0% quartile 2 10.0% 2 2.5% 0 0.5% 0 0.0% minimum 0

Summary Statistics

 Mean 4.44915 Std Dev 2.99425 Std Err Mean 0.137822 Upper 95% Mean 4.71997 Lower 95% Mean 4.17833 N 472 Sum 2100 Variance 8.96556 CV 67.2994 N Missing 475

There are many such data in real life in which measurement is done in counts, but for analysis they need to be considered continuous data.

In Summary, it depends a lot on what analysis you want to have for a particular data set. And we should have understanding of what affect a tool will have , if the data are considered continuous or attribute during the analysis.

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Continuous data is measured and attribute data is counted. Continuous data contains more information than attribute data. It is beneficial to have continuous data rather than to have  attribute data while doing some analysis like in control charts, . Continuous data can be measured, verified, and manipulated. it is known as numerical data. for example height, weight, temperature, volume, humidity etc.

Attribute data is categorical and count data. They have only a finite number of points that can be represented by non negative integers. Continuous data results fro  measurement on a continuous scale such as length, weight. these scales are called continuous because between any two values are an infinite number of other values.

There is still confusion in some specific dataset whether it should be considered as continuous or attribute. For example

1. while measuring time we think that it is hour, minutes, month, year, so it will be attribute data but in actually it is continuous data because it can be break into minutes, seconds, pico seconds like that. but we can convert it into attribute data.

2. Another confusing dataset when we are talking about money, when we withdraw money from atm it comes like 100,200,500,2000 but if we see our bank balance in our records online then we see the figures like 200.35 rupees, so it is continuous data because it has several infinite values so it is continuous data.

3. Another confusing dataset is percentage data or we can say that derived data, how we will consider it continuous or attribute. it depends on source data what it is actually according to that % data is decided.

for example  number of students taking this class divided by total number of graduate students is attribute data because a student can not be 1.5,2.5

4. Another confusion about measuring in this example, imagine a young child is sick, as apparent first thing we do is to touch his forehead to feel if it is warm or not then it is attribute data but if we measure it by thermometer then it is continuous data.

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Data is Continuous if there are infinite number of values possible during measurement.

If the data is Continuous – We need to ask ourselves if it is possible for the data to take on values that are fractions or decimals. If your answer is Yes, It is a Continuous Data. E.g. Height, Weight, Login Hours TAT etc.

Attribute/Discrete Data usually occurs in a case where there is only a certain number of values, or when we are counting something (Using Whole Numbers).

A Data type is attribute, if there are only a finite number of values possible.

E.g. True, False, On, Off etc.

Attached are some examples: More examples where a Metric can be considered both Discreet and Continuous are below:

Accuracy % : Continuous   Met/Not met : Attribute or Discreet Data

Success % : Continuous    Pass/Fail : This defines count of passes and failures.

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I am skipping the basic definition for Continuous and Attribute data, since all Excellence Ambassadors will be very knowledgeable on those. The following are 4 situations that I would like to discuss, where there is likely to be debate on the data type classification. It is to be noted that the basic nature of the metric alone may not be sufficient to determine the data type, but the assessment methodology also matters. The example of the diameter, when measured using a micrometer will be a variable data output, whereas when screened using a plug gage, becomes an attribute data output.

However, the debates do continue on the other examples, where people do get confused when they see numbers with decimals, which they tend to associate with a variable type of data.

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Data, the source and need of all investigations in an analysis has been classified based on the attributes and values it describes.

Qualitative data are the ones which focus on description, observation and attribution. Its aesthetic and difficult to measure. NOMINAL( names, labels, not ordered) and ORDINAL(ordered, attitudes) are the data levels categorized in this type.

Quantitative data are the ones which are measurable or countable. Discrete and Continuous are the classifications in it and supports all the four data levels of being NOMINAL(names, labels, not ordered), ORDINAL(ordered,attitudes), INTERVAL(ordered, no natural zero) and RATIO(ordered, with natural zero).

After establishing the measurement to gauge the data and disclose a pattern, one opts for DISCRETE or CONTINOUS form of sampling.

Discrete data has values that are distinct, countable, and separate. Bar Graphs are used in this case to represent the data.

Number of employees in an organisation

Number of defects identified

Customer surveys scores

Continuous data has values of a continuous range that can be measured, counted and ordered. Histogram and line graph are used to represent the data.

Length of road laid up per day

Amount of milk filled in the packets during packing

For instance when a manufacturing set up is to determine the efficiency of the process, the major factors to be measured are the Quality (conformance% or defect count), Productivity( utilization) and Availability (on/off time).  Quality can be QC score (continuous) or DPMO(Discrete), Productivity measured through login times(continuous) and Availability gauged through login days (discrete).

Continuous data has a distinctive edge over discrete data and if the measurement allows, one can wish for the continuum with the data. Continuous data provides better "see through" for comparatively smaller sizes of sample and is highly sensitive(probability of detection). The prediction from the data is relatively better than with discrete inputs.

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CSAT is discrete always confused being continuous data

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The confusion about continuous data and attribute data is the percentage data. In true sense, percentage data is discrete because the underlying data that the percentages are calculated from is discrete. For example: the percentage of defects is calculated by dividing the number of defects(discrete count data). In practice, percentage data are often treated as continuous because the percentage can take on any value along the continuum from 0 to 100%. Adding to it, dividing a percentage point into two or more parts.

Discrete data are easy to collect and interpret.

Continuous data senses variation. For example: Speeding of car at highway with a speed limit of 70 miles per hour. If we collect continuous data we have more information. If we use discrete data, we only know whether someone was speeding over the speed limit of 70 mph or not speeding i.e at or under 70 mph. For Example: knowing that travelling 70 mph gives a different understanding of their speed than knowing that they were travelling 90 miles per hour, even though both would be classified as speeding using is discrete data.

It is advisable to collect continuous data in practice and convert it into discrete as per the threshold value. In the above mentioned example, we would collect: Continuous Data- how fast was the automobile travelling in miles per hour.

Later determine whether the result is speeding or not speeding by comparing the actual speed to the threshold of 70 miles per hour

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Continuous data: Continuous data is information that can be measured on a scale or having limit for measurement. It can be measured and broken down into smaller parts and still have meaning.

Attribute data: Attribute data, also known as discrete data, are counted in whole numbers or integers. The result will always be a whole number—never a decimal fraction. It is also known as count data.

Any type of data collected is for analysis

Examples for confusing datasets

1.     Sizes of clothes 24,28,30,32,34,36,38,40,42,44 etc. No. of clothes available in the particular size is attribute data and measuring 10 pairs of clothes and they should have size ranging from 24-32 is continuous data.

2.     Age of children in group I is 4,2,8,5,6,4,7,2,3,6,4,7,5,3,5,1,2,9,7. This is attribute data. If the same data is having a limit of age group and the number falling into that is continuous data.

3.     No. of links in a chain is 100,102,105,100,101,103,108,105,106,102,105,107 is attribute data. Different lengths of chains is a continuous data.

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Some examples that I could think of are as follows:

1.       Delivery time in days: If you are tracking actual times (4.321 days for example) then it is continuous….

If you count entire days (4 days) then it is discrete….

2.       Height: If measuring actual height of people then it is continuous. But if we are categorizing as small, medium, tall the it become discrete data.

3.       Calls coming into the Contact Center on a daily basis: Though Call count of each day is discrete, if there are a large numbers of calls over the days then it can assumed to be a continuous data.

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The following data sets are confusing for continuous or Discrete like Age , Income, Percentage

data, Time.

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Discrete data means that the values can be taken only from a finite set. For e.g. while marking students in an exam, examiners usually give a whole mark or half a mark. Thus for a 100 mark exam, the set of possible values are 0, 0.5, 1, 1.5 ..... 99, 99.5, 100. While this would fit the definition of discrete data, practically we can consider it as a continuous data set. As a rule of thumb, if the number of distinct elements in the set exceeds 10, the data set can be considered as a continuous data set.

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Time data set is one of the confusing.. if we are having the time data set with hours, minutes and seconds - this can be classified as continuous.. if it is rounded off to nearest hour, it can be classified as discrete.

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Data are classified into two types – attribute data and measurement data

Measurement Data:  It is otherwise known as Continuous data or variables data and is obtained from the measurement taken on an item or a human being. For instance, weight and height of a person, body temperature of a person are all good examples of the continuous data.  Here data is measured on a continuous scale.

Attribute Data :

This can occur when a variable is classified either as categories or as a count data (occurrence of an event).     When it comes to categories, the categories can have a binary option or multiple options. In other words, we talk about nominal and ordinal data.  Nominal data can be for instance- passed or failed, yes/no. In an ordinal data, say, a Customer Satisfaction Survey, the categories would have a ranking, such as 1-Excellent, 2-Very Good, 3- Good, 4-Satisfactory, 5-Bad, 6-Worse.

Cases where datasets can be confusing

1.       Sometimes, the measurement system of the process that we work on, may not be good enough or calibrated to our needs.

Eg:   We want to buy vegetables in a small vegetable shop in a village.  The shop uses a weighing balance (traditional weighing method using measured stone weights) to measure the weight of vegetables.  The same quantity of vegetables bought in a Super market, might give a different value, because of different measuring unit (electronic weighing machine).

More chances are that i would treat this data (from Super market), if collected for multiple customers, as a continuous data, rather than the weight obtained from the shop in the village, which i would treat as an attribute data. The reason being, the electronic weighing machine would provide me the weight in decimals and i may have opportunities to have continuous data .

Conclusion

Thus we see there are scenarios, where your dataset can be treated as either continuous or attribute . Ultimately , the more continuous values we have, better would be trap the variation in the system and address it and ensure better process control is there

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Some of the examples of continuous or discrete data which are confusing :-

1.Rainfall :-

A variable whose values are whole numbers (counts) is called discrete. For example, the number of days with rain in a year is discrete. A variable that may contain any value within some range is called continuous. For example the total annual rainfall is continuous.

2. Time - Continuous or Discrete ?

3. Age - discrete or continuous?

Age is continuous numerical variable. (Although the ages are whole numbers, the concept of age is continuous.)

But Number of children is a discrete numerical variable .

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It would be possible to make data traditionally considered as continuous appear as attribute by an appropriately worded question. Some examples of the same are provided below. The column on the left has a set of parameters traditionally considered as continuous, while the column on the right has certain questions, which if answered will lead to “counting” of the value of the supposedly continuous variable.

 Weight of an object How many grams of matter are there in that object? Volume of water in a container How many millilitres of space or occupied by water in that container? Bank Balance How many paise are there in that account? Height of a building How many metres of height are there in that building?

If there is an argument that the value cannot be always counted in whole number of grams or ml and so on as they can be fractions thereof, the counter to that would be to narrow down the Unit of Measurement to micro-grams, pico-grams, atto-grams, femto-grams and so on and at some point of time, the value of the parameter can be counted.

To resolve the above, one just needs to stick to the unit of measurement traditionally used. For example, bank balances are normally measured in Rupees and Paise, which sustains the continuous nature of the parameter. Similarly the traditional units for weights and heights can be considered, as again the continuous nature of the parameter being measured is retained.

Some discrete data like errors can get a continuous "make-up" when averaged. For example, errors or error transactions assessed every hour are discrete but when averaged hourly over a day appears continuous.

Further, using the discrete data, “errors”, various related parameters can be derived which can appear both continuous and discrete. For example, a defect rate of 10% which is traditionally considered attribute can be also expressed as “Average defects per product” of 0.1, which would appear continuous.

Such data could perhaps be considered, "Quasi-continuous".

Additionally, in hard-core Mathematics, mixed random variables and topological sets are conceptually considered neither continuous nor discrete.

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Types of Data

There are two types of data.

Qualitative and Quantitative.

Qualitative data cannotmbe measured objectively.it can be ranked or ordered.

Quantitative data can be measured objectively.

Qualitative data is further divided into nominal,ordial and binary.

Quantitative data is divided into continous and discrete..

Most of thw cases when data is in % then the confusion starts.

The golden rule states that

1) Discrete numerator, continuos denominator than data is continuous

2) Continuous numerator and continuos denominator than data is continuous.

3) Discrete numerator and discrete denominator than data is disxrete.

4) Continuous numerator and discrete denominator than data is continuos.

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Sometimes attribute data can actually come from continuous data. For example, if we are segregating our products as 'good' or 'bad' after actually measuring their dimensions, then though we have continuous data at the back end but our final outcome is attribute data. On the other hand, sometime discrete data may also appear as continuous data, for example take a case of customer survey at a hotel's front office, where customers are asked to rate their satisfaction level at a scale of 1 to 10 against following three quality characteristics:

1) Politeness

2) Promptness

3) Ease of Billing

In this case the actual data collected is discrete but if we take the average of each customer, the data set will look like continuous.

In first example, data is both continuous as well as attribute and both can be used for taking quality improvement action, e.g. with the attribute data we may calculate the current rejection rate and set a target to improve it but as we all know continuous data is more informative, better decisions can be taken based on continuous data and the root cause of rejection can be found.

In second example of front office survey at a hotel, its necessarily discrete data only which is giving a false appearance of continuous data when its averaged.

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Percentage is an example of a dataset which can be confused to be continuous eg. % volume or marks and discrete as well eg. % of no.of students..

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If one is interested in the number of calls per period and treat it as random variable it is of discrete nature, Always need to have per period per person defined in the data.
Lets say if the number of calls per period exceeds 10, and starts falling under normal distribution, then it will start resembling continuous data.

Each time proper analysis need to be made to bucket data under continuous or discrete data.

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Continuous data and Attribute data:

Continous data is measurable where as attribute data is countable.

continous data example:- percentage of, temperature, etc..

atribute data example:- either or, number of defectives, etc..

A proper analysis is a must to conclude the data type. Number of defectives is an attribute data type, if the same defectives needs a measurement on percentage of defectives, it becomes continuous data type.

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Thanks to all participants for making this a rich discussion. Almost all of you spelled out the correct definition of Continuous and Attribute data and provided good examples. Also some of you pinpointed the area of confusion when the metric is a mix of continuous and discrete data e.g. percentages , practical usage of data, large amounts of discrete data being used as continuous , measurement gauge used etc.

Kavitha Sundar provided a comprehensive explanation to data types with examples and inference.

Congratulations Kavitha and thanks everyone for the your inputs.

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